in the given figure `y=x^(2)+bx+c` is a quadratic polynomial which meets x-axis at A and B and y-axis at C Q is vertex of quadratic polynomial and foot of perpendicular from Q to x-axis and y-axis are P and R respectively. Then answer the following questions.
Q. If `("area of" DeltaABC)/("Area of rectangle" OPQR)=(8)/(3)` satisfies the relation `b^(2)=kc` then `k` will be

Options

A. `9`
B. `(2)/(9)`
C. `(9)/(2)`
D. `2`

Answer

B

Explanation

`((1)/(2)(beta-alpha)c)/(((b^(2)-4c)/(4))(-(b)/(2)))=(8)/(3)` squaring and using
`(beta-alpha)^(2)=b^(2)-4c`
`9c^(2)=4b^(4)-16b^(2)c`
`4b^(4)-16b^(2)c-9c^(2)=0`
`4b^(4)-18b^(2)c+2b^(2)c-9c^(2)=0`
`(2b^(2)+c)(2b^(2)-9c)=0impliescgt0because2b^(2)+cne0`
`(b^(2))/(c)=(9)/(2)implies(b^(2))/(c)=4.5`